cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-28 of 28 results.

A113857 a(n) = binomial(4+2*n, n) * binomial(9+2*n, 4+n).

Original entry on oeis.org

126, 2772, 48048, 772200, 12033450, 184940756, 2824549728, 43028530272, 655081791000, 9977399586000, 152112583402560, 2322021633001200, 35496198345658050, 543418421128852500, 8331507823355640000, 127919340117331963200, 1966759854303978934200, 30279186980267369086800
Offset: 0

Views

Author

Zerinvary Lajos, Feb 02 2006

Keywords

Comments

If one uses the "table" view of array A062190, the sequence appears as the fourth column right from the middle in the "formatted as a triangular array" subpanel.

Examples

			a(0) = C(4+2*n,n)*C(9+2*n,4+n) = C(4,0)*C(9,4) = 1*126 = 126.
a(7) = C(4+2*7,7)*C(9+2*7,4+7) = C(18,7)*C(23,11) = 31824*1352078 = 43028530272.

Links

Crossrefs

Cf. A001700, A002694, A062190.

Programs

  • Mathematica
    a[n_] := Binomial[4+2*n, n] * Binomial[9+2*n, 4+n]; Array[a, 20, 0] (* Amiram Eldar, Sep 05 2025 *)
  • PARI
    a(n)={binomial(4+2*n, n) * binomial(9+2*n, 4+n)} \\ Andrew Howroyd, Jan 07 2020

Formula

a(n) = A062190(4+2*n, 4+n).
a(n) = A002694(n+2)*A001700(n+4). - R. J. Mathar, Nov 28 2008
a(n) ~ 2^(4*n+13) / (Pi*n). - Amiram Eldar, Sep 05 2025

Extensions

Definition rephrased by R. J. Mathar, Nov 28 2008
Edited and more terms added by Andrew Howroyd, Jan 07 2020

A113895 a(n) = C(2+2*n, n) * C(7+2*n, 2+n).

Original entry on oeis.org

21, 336, 4950, 72072, 1051050, 15402816, 226972746, 3362559200, 50062040028, 748664904000, 11241203533560, 169398104243760, 2561053271692500, 38833447762771200, 590405728218941250, 8998028449224091200, 137437148161776305700, 2103486475191421320000, 32253916565936980114200
Offset: 0

Views

Author

Zerinvary Lajos, Jan 28 2006

Keywords

Examples

			If n=0 then C(2+2*0,0)*C(7+2*0,2+0) = C(2,0)*C(7,2) = 1*21 = 21.
If n=7 then C(2+2*7,7)*C(7+2*7,2+7) = C(16,7)*C(21,9) = 11440*293930 = 3362559200.
If n=10 then C(2+2*10,10)*C(7+2*10,2+10) = C(22,10)*C(27,12) = 646646*17383860 = 11241203533560.

Links

Crossrefs

Cf. A001791, A002054, A062190.

Programs

  • Maple
    seq(binomial(2+2*n,n)*binomial(7+2*n,2+n), n=0..20); # Robert Israel, Dec 16 2018
  • Mathematica
    nmax = 10; NN[5, m_, x_] := x^m*(2*m+5)!*Hypergeometric2F1[-m, -m, -2*m-5, (x-1)/x]/((m+5)!*m!); tri = Table[CoefficientList[NN[5, m, x], x], {m, 0, 2*nmax+2}]; Table[tri[[2n+3, n+3]], {n, 0, nmax}] (* Jean-François Alcover, Sep 18 2013 *)
  • PARI
    a(n) = binomial(2+2*n,n)*binomial(7+2*n,2+n); \\ Michel Marcus, Sep 18 2013

Formula

a(n) = C(2+2*n, n) * C(7+2*n, 2+n).
From Robert Israel, Dec 16 2018: (Start)
a(n) = A001791(n+1)*A002054(n+3).
24*(11+2*n)*(5+2*n)*(n+3)*a(n+1) - 2*(n+4)*(11*n^2+112*n+252)*a(n+2)+(8+n)*(n+5)*(n+3)*a(n+3) = 0. (End)
a(n) ~ 2^(4*n+9) / (Pi*n). - Amiram Eldar, Sep 06 2025

Extensions

Simpler name and more terms added by Joerg Arndt, Sep 18 2013

A114059 a(n) = binomial(3+2*n, n) * binomial(8+2*n, 3+n).

Original entry on oeis.org

56, 1050, 16632, 252252, 3775200, 56316546, 840639800, 12575971408, 188663555808, 2838687761000, 42836302222560, 648207031545000, 9834444563299200, 149569451148798450, 2279905857066915000, 34825702701626575200, 532997250488883180000, 8172044956118671828200
Offset: 0

Views

Author

Zerinvary Lajos, Feb 02 2006

Keywords

Examples

			a(0) = C(3+2*0,0)*C(8+2*0,3+0) = C(3,0)*C(8,3) = 1*56 = 56.
a(10) = C(3+2*10,10)*C(8+2*10,3+10) = C(23,10)*C(28,13) = 1144066*37442160 = 42836302222560.

Links

Crossrefs

Cf. A062190.

Programs

  • Mathematica
    a[n_] := Binomial[3+2*n, n] * Binomial[8+2*n, 3+n]; Array[a, 20, 0] (* Amiram Eldar, Sep 05 2025 *)
  • PARI
    a(n)={binomial(3+2*n, n) * binomial(8+2*n, 3+n)} \\ Andrew Howroyd, Jan 07 2020

Formula

a(n) = A062190(3+2*n, 3+n).
a(n) ~ 2^(4*n+11) / (Pi*n). - Amiram Eldar, Sep 05 2025

Extensions

Name edited and terms a(11) and beyond from Andrew Howroyd, Jan 07 2020

A114238 a(n) = binomial(2+2*n, 2+n) * binomial(7+2*n, n).

Original entry on oeis.org

1, 36, 825, 16016, 286650, 4900896, 81477396, 1330243200, 21455160012, 343138081000, 5455289950110, 86359817849760, 1362899694292500, 21460589553110400, 337374701839395000, 5297540558850547200, 83114164698623615700, 1303247055281641470000, 20427480491760087405660
Offset: 0

Views

Author

Zerinvary Lajos, Feb 04 2006

Keywords

Examples

			For n=0 then C(2+2*0,2+0)*C(7+2*0,0+0) = C(2,2)*C(7,0) = 1*1 = 1.
For n=8 then C(2+2*8,2+8)*C(7+2*8,0+8) = C(18,10)*C(23,8) = 43758*490314 = 21455160012.

Links

Crossrefs

Cf. A062190.

Programs

  • Mathematica
    a[n_] := Binomial[2*n+2, 2+n] * Binomial[2*n+7, n]; Array[a, 25, 0] (* Amiram Eldar, Sep 05 2025 *)
  • PARI
    a(n)={binomial(2+2*n, 2+n) * binomial(7+2*n, 0+n)} \\ Andrew Howroyd, Nov 08 2019

Formula

a(n) ~ 2^(4*n+9) / (Pi*n). - Amiram Eldar, Sep 05 2025

Extensions

Terms a(13) and beyond from Andrew Howroyd, Nov 08 2019

A114253 a(n) = C(5+2*n,5+n)*C(10+2*n,0+n).

Original entry on oeis.org

1, 84, 3276, 92400, 2187900, 46558512, 923410488, 17439488352, 317907339750, 5644249611000, 98209943231400, 1682207622669600, 28457345616827400, 476607460678020000, 7917519856977720000, 130649634333275016960, 2143941655711783421340, 35018537985874435552560
Offset: 0

Views

Author

Zerinvary Lajos, Feb 04 2006

Keywords

Examples

			If n=1 then C(5+2*1,5+1)*C(10+2*1,0+1) = C(7,6)*C(12,1) = 7*12 = 84.
If n=11 then C(5+2*n,5+n)*C(10+2*n,0+n) = C(27,16)*C(32,11) = 13037895*129024480 = 1682207622669600.

Links

Crossrefs

Cf. A003516, A004311, A062190.

Programs

  • Maple
    seq(binomial(5+2*n,5+n)*binomial(10+2*n,n),n=0..30); # Robert Israel, Jan 11 2019
  • Mathematica
    a[n_] := Binomial[2*n + 5, n + 5]*Binomial[2*n + 10, n]; Array[a, 20, 0] (* Amiram Eldar, Sep 06 2025 *)

Formula

From Robert Israel, Jan 11 2019: (Start)
(n+1)^2*(11+n)*a(n+1) = 4*(7+2*n)*(3+n)*(11+2*n)*a(n).
a(n) ~ 32768*16^n/(Pi*n). (End)
a(n) = A003516(n+2) * A004311(n+5). - Amiram Eldar, Sep 06 2025

A113888 a(n) = C(2*n+1,n)*C(2*n+6,n+1).

Original entry on oeis.org

6, 84, 1200, 17325, 252252, 3699696, 54609984, 810616950, 12092280200, 181176906768, 2725140250560, 41132585656890, 622787147955000, 9456196695480000, 143946539451475200, 2196309308974461450, 33581927605139911800, 514470608092210770000, 7895695609776494520000
Offset: 0

Views

Author

Zerinvary Lajos, Jan 28 2006

Keywords

Examples

			If n=0 then C(1+2*0,0+0)*C(6+2*0,1+0) = C(1,0)*C(6,1) = 1*6 = 6.
If n=4 then C(1+2*4,0+4)*C(6+2*4,1+4) = C(9,4)*C(14,5) = 126*2002 = 252252.
If n=10 then C(1+2*10,0+10)*C(6+2*10,1+10) = C(21,10)*C(26,11) = 352716*7726160 = 2725140250560.

Crossrefs

Cf. A001700, A002694, A062190.

Programs

  • Mathematica
    Table[Binomial[2n+1,n]Binomial[2n+6,n+1],{n,0,20}] (* Harvey P. Dale, Jun 14 2011 *)

Formula

From Amiram Eldar, Sep 06 2025: (Start)
a(n) = A001700(n) * A002694(n+3).
a(n) ~ 2^(4*n+7) / (Pi*n). (End)

Extensions

Edited by N. J. A. Sloane, Feb 03 2007

A114251 a(n) = C(3+2*n,3+n)*C(8+2*n,0+n).

Original entry on oeis.org

1, 50, 1386, 30576, 600600, 11027016, 193993800, 3316739712, 55588369122, 918398981500, 15013703965260, 243495727020000, 3925151119562400, 62976611010020400, 1006711677146430000, 16046173576808179200, 255175067057177767500, 4050491847815341688760
Offset: 0

Views

Author

Zerinvary Lajos, Feb 04 2006

Keywords

Examples

			If n=0 then C(3+2*0,3+0)*C(8+2*0,0+0) = C(3,3)*C(8,0) = 1*1 = 1.
If n=7 then C(3+2*7,3+7)*C(8+2*7,0+7) = C(17,10)*C(22,7) = 19448*170544 = 3316739712.

Crossrefs

Cf. A002054, A004310, A062190.

Programs

  • Mathematica
    Table[Binomial[3+2n,3+n]Binomial[8+2n,n],{n,0,20}] (* Harvey P. Dale, Jul 26 2019 *)

Formula

From Amiram Eldar, Sep 06 2025: (Start)
a(n) = A002054(n+1) * A004310(n+4).
a(n) ~ 2^(4*n+11) / (Pi*n). (End)

Extensions

More terms from Harvey P. Dale, Jul 26 2019

A114252 a(n) = C(4+2*n,4+n)*C(9+2*n,0+n).

Original entry on oeis.org

1, 66, 2184, 54600, 1178100, 23279256, 434546112, 7801876368, 136246002750, 2331320491500, 39283977292560, 654191853260400, 10794165578796600, 176805993477330000, 2879098129810080000, 46660583690455363200, 753276797952788769660, 12121801610494996922040
Offset: 0

Views

Author

Zerinvary Lajos, Feb 04 2006

Keywords

Crossrefs

Cf. A002694, A030054, A062190.

Programs

  • Mathematica
    Table[Binomial[4+2n,4+n]Binomial[9+2n,n],{n,0,20}] (* Harvey P. Dale, Aug 30 2015 *)

Formula

From Amiram Eldar, Sep 06 2025: (Start)
a(n) = A002694(n+2) * A030054(n+4).
a(n) ~ 2^(4*n+13) / (Pi*n). (End)

Extensions

More terms from Harvey P. Dale, Aug 30 2015
Previous Showing 21-28 of 28 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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